Step 1: Analyze the expression.
The given statement is:
\[
(p \land \sim q) \lor ((\sim p) \land q) \lor ((\sim p) \land (\sim q))
\]
This expression represents a combination of three conditions. We will break it down logically to see if it can be simplified.
Step 2: Breakdown of each term.
1. \( p \land \sim q \): This term represents the case where \( p \) is true and \( q \) is false.
2. \( (\sim p) \land q \): This term represents the case where \( p \) is false and \( q \) is true.
3. \( (\sim p) \land (\sim q) \): This term represents the case where both \( p \) and \( q \) are false.
Step 3: Examine the entire disjunction.
The disjunction \( (p \land \sim q) \lor ((\sim p) \land q) \lor ((\sim p) \land (\sim q)) \) covers all cases where:
\( p \) is true and \( q \) is false,
\( p \) is false and \( q \) is true,
both \( p \) and \( q \) are false.
We can see that the only case not covered is where both \( p \) and \( q \) are true.
Step 4: Simplification using logical laws.
We can use the distributive property to simplify the expression. First, notice that \( (p \land \sim q) \lor ((\sim p) \land q) \) simplifies to \( p \lor q \), because it covers all cases except when both \( p \) and \( q \) are true. Now, we add \( ((\sim p) \land (\sim q)) \), which means that either \( p \) or \( q \) can be false, covering all cases except when both \( p \) and \( q \) are true.
Thus, the entire expression simplifies to \( \sim p \lor \sim q \).
Step 5: Conclusion.
The given logical expression is equivalent to \( \sim p \lor \sim q \).